This is a table of Clebsch–Gordan coefficients used for adding angular momentum values in quantum mechanics. The overall sign of the coefficients for each set of constant j 1 , j 2 , j is arbitrary to some degree and has been fixed according to the Condon-Shortley and Wigner sign convention as discussed by Baird and Biedenharn. Tables with the same sign convention may be found in the Particle Data Group's Review of Particle Properties and in online tables.
The Clebsch–Gordan coefficients are the solutions to
| ( j 1 j 2 ) j m ⟩ = ∑ m 1 = − j 1 j 1 ∑ m 2 = − j 2 j 2 | j 1 m 1 j 2 m 2 ⟩ ⟨ j 1 j 2 ; m 1 m 2 | j 1 j 2 ; j m ⟩ Explicitly:
⟨ j 1 j 2 ; m 1 m 2 | j 1 j 2 ; j m ⟩ = δ m , m 1 + m 2 ( 2 j + 1 ) ( j + j 1 − j 2 ) ! ( j − j 1 + j 2 ) ! ( j 1 + j 2 − j ) ! ( j 1 + j 2 + j + 1 ) ! × ( j + m ) ! ( j − m ) ! ( j 1 − m 1 ) ! ( j 1 + m 1 ) ! ( j 2 − m 2 ) ! ( j 2 + m 2 ) ! × ∑ k ( − 1 ) k k ! ( j 1 + j 2 − j − k ) ! ( j 1 − m 1 − k ) ! ( j 2 + m 2 − k ) ! ( j − j 2 + m 1 + k ) ! ( j − j 1 − m 2 + k ) ! . The summation is extended over all integer k for which the argument of every factorial is nonnegative.
For brevity, solutions with m < 0 and j1 < j2 are omitted. They may be calculated using the simple relations
⟨ j 1 j 2 ; m 1 m 2 | j 1 j 2 ; j m ⟩ = ( − 1 ) j − j 1 − j 2 ⟨ j 1 j 2 ; − m 1 , − m 2 | j 1 j 2 ; j , − m ⟩ .
and
⟨ j 1 j 2 ; m 1 m 2 | j 1 j 2 ; j m ⟩ = ( − 1 ) j − j 1 − j 2 ⟨ j 2 j 1 ; m 2 m 1 | j 2 j 1 ; j m ⟩ .
When j2 = 0, the Clebsch–Gordan coefficients are given by δ j , j 1 δ m , m 1 .
Algorithms to produce Clebsch–Gordan coefficients for higher values of j 1 and j 2 , or for the su(N) algebra instead of su(2), are known. A web interface for tabulating SU(N) Clebsch-Gordan coefficients is readily available.
